Percentages
Percentages are used in everyday life, for example, calculating discounts during sales and interest rates at banks. Knowing how to find and use percentages is an important skill.
Calculating a percentage of a quantity
A percentage is a way of expressing a fraction or a decimal as parts of 100.
The symbol '%' means 'per cent'. 9% means 9 out of every 100.
9% = 0.9 = \(\frac{9}{10}\)
Knowledge of converting between decimals, fractions and percentages is required to complete this section. You can revise this here if you need to.
To find a percentage of a quantity
- First find 1% by dividing by 100
- Then multiply by the percentage that you want.
Example
Angela works part time and gives 3% of her earnings to charity. One week she earned ÂŁ275.
How much did she give to charity that week?
Solution
First, find 1% by dividing by 100.
ÂŁ275 Ă· 100 = ÂŁ2.75
Then multiply by the percentage that you want.
*ÂŁ2.75 x 3 = ÂŁ8.25
Answer
Angela gave ÂŁ8.25 to charity that week.
Question
A golf club has 550 members. 32 percent of them are female.
How many female members does the golf club have?
Solution
First find 1% by dividing by 100.
Then multiply by the percentage that you want.
550 Ă·100 x 32 = 176
We can do both in one calculation.
Answer:
There are 176 female members in the golf club.
Writing one quantity as a percentage of another
Sometimes we may want to describe one number as a percentage of another. For example, a teacher marking a test with a maximum of 40 marks will probably want to write each pupil’s score as a percentage rather than a mark out of 40.
To write one quantity as a percentage of another:
- Divide one quantity by the other
- Then multiply by 100.
Example
Ryan got 52 out of 65 in his class test. Simon got 72% in the same test.Who has done better in the test and by what percentage?
Solution:
Change Ryan’s score to a percentage:
- Divide one quantity by the other
- Then multiply by 100.
52 Ă· 65 x 100 = 80
Ryan got 80% in his test and Simon got 72%.
Answer:
Ryan did better by 8%
Question
In a box of 250 apples, 18 have gone bad. What percentage is this?
Solution
- Divide one quantity by the other
- Then multiply by 100.
18 Ă· 250 x 100 = 7.2
We can do both in one calculation.
Answer:
7.2% of the apples have gone bad.
Increasing and decreasing an amount by a percentage
Example
A car dealership aims to increase their sales by 15% over the next year. If they have sold 720 cars this year, how many do they need to sell next year to meet their target?
Solution:
First, work out 15% of 720.
720 Ă· 100 x 15 = 108
The dealership needs to sell 108 more cars. To calculate how many cars they need to sell next year, add 108 to the number they sold this year.
720 + 108 = 828
Answer:
They need to sell 828 cars to meet their target.
Example
A Grand Slam tennis racquet normally costs ÂŁ125. In a sale, the price is reduced by 30%. What is the sale price of the racquet?
Solution:
First, work out 30% of ÂŁ125.
125 Ă· 100 x 30 = 37.5
The price has been reduced by ÂŁ37.50.
To calculate the sale price, subtract ÂŁ37.50 from the original price.
125 – 37.50 = £87.50
Answer:
The sale price of the Grand Slam racquet is ÂŁ87.50.
Question
In 2013, Lake Poyang in China covered an area of 1340 square kilometres. In 2023, it was estimated that the area had shrunk by 22%.
What area did Lake Poyang cover in 2023? Give your answer to the nearest square kilometer.
Solution:
First, find 22% of 1340.
1340 Ă· 100 x 22 = 294.8
The lake is 294.8 sq km smaller.
1340 – 294.8 = 1045.2 (1045 to the nearest sq km).
Answer:
The lake covered an area of 1045 sq km in 2023.
Percentage increase and decrease
When a value has been increased or decreased, the amount of change can be written as a percentage of the original value.
\(\Large \text{percentage increase} = \frac{\text{actual increase}}{\text{original value}} \times {100}\)
\(\Large \text{percentage decrease} = \frac{\text{actual decrease}}{\text{original value}} \times {100}\)
Example
In June, Sam’s ice cream van sold 5580 vanilla cones. In July, 5859 were sold. What was the percentage increase in sales?
Solution
First work out the actual increase in sales.
5859 – 5580 = 279
Then use the formula for % increase.
\(\color{purple} {\text{percentage increase}} = \frac{\text{actual increase}}{\text{original value}} \times {100}\)
\(\frac{279}{5580} \times {100} = 5 \)
Answer:
Sales have increased by 5% from June to July.
Example
A fizzy drink contains 45 grams of sugar. The manufacturers reduce this amount to 39 grams. What is the percentage decrease in sugar content? Give your answer to the nearest percent.
Solution
First work out the actual decrease in sugar content.
45 – 39 = 6
Then use the formula for % decrease.
\(\color{purple} {\text{percentage decrease}} = \frac{\text{actual decrease}}{\text{original value}} \times {100}\)
\(\frac{6}{45} \times {100} = 13.3333333…\)
Answer:
The sugar content has been decreased by 13%.
Question
The population of a town increased from 61400 in 2021 to 66312 in 2022. What was the percentage increase in the population over this time?
Solution
First work out the actual increase in the population.
66312 – 61400 = 4912
Then use the formula for % increase.
\(\color{purple} {\text{percentage increase}} = \frac{\text{actual increase}}{\text{original value}} \times {100}\)
\(\frac{4912}{61400} \times {100} = 8 \)
Answer:
The population of the town has increased by 8%.
Test yourself
More on M1: Number
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